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Yes sharing setting for account is public read write. if do as "private" community user is not able to see other records. But I want cummunity user to edit account information if account owner is system admin.
You need to share the accounts with community user instead of making it public read write. Try sharing rules or create a Visualforce page that checks the user and updates account without sharing.
Visualforce page is not possible. How can I do with sharing rules. Should org wide setting for account be private? What should be "Select which records to be shared" and "Select the users to share with"?
As per your comment you want to share accounts owned by sys admin ... you can share it with community users by adding them to group or based on their role. If you try it, you will be able to figure it out.
Can you please elaborate how can this be done. For user system admin I have given COO role. Also created a public group with portal role. How will be the account sharing rule for account? How user will not see other accounts and see only his and will be able to edit account owned by system admin?